Dr. Kim encouraged us to reflect on the following question after class: What makes Simon's (1995) theory constructivist? (NF 2/17)
Just to give more detail to the question above: When we plan our lessons as teachers, we usually take into account the possible tendencies (e.g., actions, thinking, misconceptions) of our students, even if we do not adopt a constructivist perspective. In light of this, what makes Simon’s (1995) model of teaching constructivist? What qualities of his theory make Simon’s model “constructivist”?
Steffe and D' Ambrosio (1995) asserted that Simon may have *incorrectly* assumed that his students had weak knowledge of multiplication and that he should have activated students' prior knowledge in order for them to successfully do the task. However, Simon (1995) responded that activating prior learning may lead to knowledge that is less extensive. So when is "activating prior learning" as described by Steffe and D' Ambrosio appropriate? When is it not? (JH, 2/15)
I had to similar question to that asked by Janice (directly above). In Laura's class in the fall of 2008, one idea that came up was that of "didactic contract" (Brousseau is quoted in the Boaler, Ball, and Evan piece). Brousseau's concern was that teachers and students had a kind of tacit agreement to lower the cognitive demands of tasks, perhaps through leading questions on the part of the teacher. How do we distinguish between activity in the zone of proximal development (guided by a more capable peer or adult) or activating prior learning as in Steffe and D'Ambrosio and didactic contract? (RK)
According to Confrey (1990), on p. 111 powerful students’ constructions have those characteristics listed. Who decides whether the students’ construction is powerful or not? Must all ten attributes be satisfied for the construction to be powerful? How would the student verify historic continuity, guide for future action etc? (NA)
-NA- Confrey offers these ten items as an illustration of what powerful constructions are characterized by. He also mentions that this list is not exhaustive - this means that these ten characteristics cannot by themselves (and without anything else) decide whether a construction is powerful. I think he is simply saying that given a group of powerful constructions (that I assume the field agrees is powerful), these are some of their characteristics. (JE, 2/16)
What should teachers do when students’ understanding of the material is really weak and sketchy (Confrey, 1990; p. 121)? (NA)
-NA- I might rephrase this given the chapter title: what might the implications be for the constructivist teacher when face with students whose understanding of the material is "weak and sketcy?" (JE, 2/16)
Is the zone of potential construction in Simon’s hypothetical learning trajectory proposed by (Steffe and D’Ambrosio, 1995) synonymous to Vygotsky’s zone of Proximal development? (NA)
-NA- I think the phrase "zone of potential construction" actually comes from the Steffe and D'Ambrosio (1995) piece (see. p. 154). They claim that the "zpc" would "fit somewhere inside Simon's HLT" and that it was "implicit" (p. 154). That said, I also think that this phrase is not synonymous with Vygotsky's ZPD, but more of a modeling of terminology. (JE, 2/16)
What is the learning paradox? (NA)
As a partial answer to that question, I would point to Simon's response, in which he mentions making "the role of the mathematics teacher problematic" (p. 162). I see this as a call for teachers to be actively intellectually engaged in their practice (I feel there is additional support for this throughout the readings for the last week). I also feel it calls for a certain amount of teacher knowledge (e.g. Ball, Thames, & Phelps, 2008; Shulman, 1987). My question is, "what can teacher educators do to foster a view of mathematics teaching as problem solving?" (RK)
- Rob, I remember having a book in Math 6530 that might address this question partially---Jonathan, Nicole, maybe you can type in the details for it? I don't have it on my Endnote. (JH)
The position Simon (1995) takes on constructivism is one of social constructivism (see p. 117). In his later work (Simon et al., 2004), the work is done from a radical constructivist approach (see p. 306). Do the studies done in each case actually follow the approach to constructivism intended? Are the studies done from the same constructivist approach? Why the difference in constructivist approaches by the same author? (JE, 2/15)
How well does the model of Confrey (1990) fit Simon's claim that constructivism "provides a useful framework for thinking about mathematics learning in classrooms...[but] it does not stipulate a particular model" (p.114)? (JE, 2/16)
How do we teach students to use the process of reflective abstraction (Simon, et al., 2004; Dubinksy, 1991) or the reflective process (Confrey, 1990)? (JE, 2/16)
Adding to Jonathon's question above, how do we guide future teachers this reflective process? How can we equip them to reflect on their own teaching and learning? Especially in light of the fact that they are primarily learning through the lecture method, how best can we introduce them to constructivist ideas without overwhelming them (and ourselves) with so many approaches? (CZ, 2/16)
I had some questions about theory building. They're not gelling well for me, but here they are. What is the desired level of specificity in a theory of teaching? This week we saw some theories that carry implications for the daily practice of teaching and these results do not seem content specific. I could think about HLT as I design and carry out instruction for a differential equations course or for a grade three classroom. Are there possible gains in looking at HLT within specific content domains? For example, Sfard and Linchevski (1994) applied their theory of reification to algebra. (RK)
-RK- I was thinking about similar questions of this nature. Some of the authors we have read did not do much specific work in mathematics education (such as Piaget), while others have worked extensively in it (such as von Glasersfeld). In broader terms of constructivism, I wonder whether it is used and applied in other disciplines. I can easily see the similarities between problem-based mathematics and inquiry-based science classrooms. Similarities are less clear between mathematics, and say, history or foreign languages. With the idea that constructivism is centered around the constructing of new knowledge (whether this is cognitive, ie radical constructivist, or social), the use in mathematics classrooms seems centered around problem solving mathematics. With this being the case, I feel like constructivist mathematics classrooms are built around problem solving, not direct instruction. In history or foreign language classrooms, it doesn't seem like students are "problem solving" as it were, as much as they are assimilating new knowlege/vocabulary, and using that knowledge to analyze situations historically or to communicate. Without knowing more about how constructivism is used in other disciplines all together, it seems as if if we consider constructivist approaches those that encourage students to construct their knowledge, then mathematics classrooms also contain students constructing their knowledge, albeit from a direct instruction approach. Perhaps we need to revisit in class some of the deeper implications of constructivism. (JE, 2/16)
Questions about level of specificity also come to my mind when I think about the amount of prescription in a theory. Clearly the readings do not support a theory of teaching that prescribes moment-to-moment actions by the teacher (teaching is to be problematized, and so the teacher must be left room to make some of her own decisions). But one also wouldn't want a theory of teaching that is too vague. Balancing this inherent tension seems an important part of theory-building and I wonder how this weeks theories look like through this lens. (RK)
Rob, Rob, Rob, Rob, Rob - have you heard nothing about Assessment for Learning in the past year! (friendly jab, friendly jab). In response to your "clearly argument" I would point you to Simon (1995), p. 138, where he says the following: "the teacher is continually engaged in adjusting the learning trajectory that he [she] has hypothesized to better reflect enhanced knowledge." If this does not entail "moment-to-moment" (see Classroom Assessment Minute by Minute, Day by Day, by Leahy, Lyon, Thompson, and Wiliam, 2005) actions, what does? (JE, 2/16)
JE - I was not clear, I see. By "prescribes moment-to-moment actions" I mean rigid descriptions of teacher actions that span the time from the moment the teacher walks in the door to the moment they walk out. "First, you should do this. Then this. Etc." As an example of what I do not mean, I would point to the Launch-Explore-Summarize sequence of Core-Plus. This is presription for a sequence of activities, but there is quite a bit of room for variability within each activity. The quote that you point to, in which "the teacher is continually engaged in adjusting the learning trajectory that he [she] has hypothesized to better reflect enhanced knowledge" (p. 138) says little about how the teacher is engaged in making adjustments; that is, how the teacher solves the problems that arise in the course of following the learning trajectory. This, I believe, is left largely on the level of the teacher, who is to be an active problem-solver in the classroom. Perhaps your Leahy, Lyon, Thompson, and William (2005) reference contains some suggestions that can guide this problem-solving activity, but alas I have not yet read it. (RK)
Can I ask, too, what kind of differences do we see between theories proposed by constructivists and those proposed by logical positivists? Sometimes this week I felt as though I was reading about "how students really learn" in a way that seemed almost to acknowledge the presence and influence of an external reality. How do constructivists balance their skepticism with the need to put forth "global" theories of teaching and learning? Would a constructivist also support "local" theory building (on the level of individual teachers) through the publication of qualitative studies such as narratives? (RK)
As I was reading these past three weeks about constructivism, I kept thinking back to how I learned as a student and I thought that there were some concepts that did not produce a "learning perturbation" for me such as definitions or postulates. Does this mean that these concepts are not worth teaching because it may be hard to "learn" (as I did not construct it)? Or am I thinking about this all wrong? (JH)
-JH- I think back to a situation in teaching about right triangles, and I "perturbed" the students by looking at a "right triangle" on a globe, eventually getting them to see that we can have a triangle with three 90-degree angles. I think a perturbation is a situation where a student's current thinking is jeopardized and they have to decide how to adapt that information. (JE, 2/16)
This might be related to Janice's question directly above, but What exactly does it mean to be an active learner? (See Simon, 1995, p. 118). (NF)
-NF- See Phillips (1995), p. 9, right column for a football analogy of what active learning means (I swear, I didn't know of this article when we had this discussion a few weeks ago!) (JE, 2/16) -- In general, sports analogies don't resonate with me. Maybe that's why I glossed over it the first time... NF
Simon (1995) (among others) claims that conceptual development can be theoretically explained by Piaget’s notion of reflective abstraction. In reflecting on my own teaching experiences, I have difficulties in finding experiential evidence of conceptual knowledge development occurring. Any ideas on how to determine whether “the lesson fostered the development of conceptual understanding? Simon (1995) suggests that when students overcome obstacles that this provides evidence of conceptual growth (p. 139). What else? (NF)
-NF- I think to my own learning experiences in that the "challenge to body and mind" (Simon, 1995, p. 139) spurred me to grow mathematically. In fact, most growth came about this way. In other words, without cognitive conflict (or perturbations) growth was much more difficult to come by, even in classrooms dominated by direct instruction. To me, this comes back to the adaptation (assimilation or accommodation) of Piaget. (JE, 2/16)
Might a constructivist say that all humans engage in von Glasersfeld’s first and second principles of constructivism but not all necessarily believe in this perspective? (Inspired by Steffe & D’Ambrosio, 1995, p. 146). In a related vein, must both of von Glasersfeld’s principles be satisfied in order for one to instruct from a constructivist orientation? (Principles: “knowledge is not passively received but is actively built up by the cognizing subject”; “the function of cognition is adapted and serves in the organization of the experiential world rather than in the discovery of ontological reality” (von Glasersfeld, 1989, p. 162).) (NF)
-NF- If all humans engage in von Glasersfeld's two principles, then how do we distinguish between direct instruction and the theoretical implications of constructivism to teaching (Confrey, 1990)? (JE, 2/16)
Please help me remember if we have classified Vygotsky as a constructivist or not---Phillips classified him as such (Phillips, 1995, p. 7 first column) (JH).
-JH- I think of Vygotsky as a social constructivist. (JE, 2/16)
Confrey makes the statement (pg 111) : "To a constructivist, knowledge without belief is contradictory. Thus I wish to assert that personal autonomy is the backbone of the process of construction." Could someone help me understand how the second statement follows from the first. I do not see the relationship between them. (JDS)
-JDS- Perhaps a concrete example of knowledge without belief would be beneficial. Think of an elementary student who sees two equal-sized glasses of water, and is asked if they have the same amount of liquid (which they will respond to with a yes). When the water in one glass is transferred to a glass of a different size (in front of the student), and then asked the same question, they will respond that one glass has more water in it. This is a case where they will have knowledge that conflicts with belief. Once this conflict is discovered by the teacher (call it a misconception if you will), the teacher can then work with the student toward a better understanding. Hope this helps. (JE, 2/16)
Confrey also has a list of characterisations of strong constructions but states that the list is not exhaustive. Do we think it is possible to create an exhaustive list? In other words can we come up with a rubric for our students constructions? (I realise under the assumptions of constructivism one can never assess the constructs of another directly, but just as a thought exercise, would it be possible to make a rubric that would characterise a perfect construction?) (JDS)
-JDS- In order to construct the rubric, we would first have to agree on what the exemplary (perfect) construction would look like. I think this would be a challenge. (JE, 2/16)
Just to give more detail to the question above: When we plan our lessons as teachers, we usually take into account the possible tendencies (e.g., actions, thinking, misconceptions) of our students, even if we do not adopt a constructivist perspective. In light of this, what makes Simon’s (1995) model of teaching constructivist? What qualities of his theory make Simon’s model “constructivist”?
Steffe and D' Ambrosio (1995) asserted that Simon may have *incorrectly* assumed that his students had weak knowledge of multiplication and that he should have activated students' prior knowledge in order for them to successfully do the task. However, Simon (1995) responded that activating prior learning may lead to knowledge that is less extensive. So when is "activating prior learning" as described by Steffe and D' Ambrosio appropriate? When is it not? (JH, 2/15)
I had to similar question to that asked by Janice (directly above). In Laura's class in the fall of 2008, one idea that came up was that of "didactic contract" (Brousseau is quoted in the Boaler, Ball, and Evan piece). Brousseau's concern was that teachers and students had a kind of tacit agreement to lower the cognitive demands of tasks, perhaps through leading questions on the part of the teacher. How do we distinguish between activity in the zone of proximal development (guided by a more capable peer or adult) or activating prior learning as in Steffe and D'Ambrosio and didactic contract? (RK)
According to Confrey (1990), on p. 111 powerful students’ constructions have those characteristics listed. Who decides whether the students’ construction is powerful or not? Must all ten attributes be satisfied for the construction to be powerful? How would the student verify historic continuity, guide for future action etc? (NA)
-NA- Confrey offers these ten items as an illustration of what powerful constructions are characterized by. He also mentions that this list is not exhaustive - this means that these ten characteristics cannot by themselves (and without anything else) decide whether a construction is powerful. I think he is simply saying that given a group of powerful constructions (that I assume the field agrees is powerful), these are some of their characteristics. (JE, 2/16)
What should teachers do when students’ understanding of the material is really weak and sketchy (Confrey, 1990; p. 121)? (NA)
-NA- I might rephrase this given the chapter title: what might the implications be for the constructivist teacher when face with students whose understanding of the material is "weak and sketcy?" (JE, 2/16)
Is the zone of potential construction in Simon’s hypothetical learning trajectory proposed by (Steffe and D’Ambrosio, 1995) synonymous to Vygotsky’s zone of Proximal development? (NA)
-NA- I think the phrase "zone of potential construction" actually comes from the Steffe and D'Ambrosio (1995) piece (see. p. 154). They claim that the "zpc" would "fit somewhere inside Simon's HLT" and that it was "implicit" (p. 154). That said, I also think that this phrase is not synonymous with Vygotsky's ZPD, but more of a modeling of terminology. (JE, 2/16)
What is the learning paradox? (NA)
As a partial answer to that question, I would point to Simon's response, in which he mentions making "the role of the mathematics teacher problematic" (p. 162). I see this as a call for teachers to be actively intellectually engaged in their practice (I feel there is additional support for this throughout the readings for the last week). I also feel it calls for a certain amount of teacher knowledge (e.g. Ball, Thames, & Phelps, 2008; Shulman, 1987). My question is, "what can teacher educators do to foster a view of mathematics teaching as problem solving?" (RK)
- Rob, I remember having a book in Math 6530 that might address this question partially---Jonathan, Nicole, maybe you can type in the details for it? I don't have it on my Endnote. (JH)
The position Simon (1995) takes on constructivism is one of social constructivism (see p. 117). In his later work (Simon et al., 2004), the work is done from a radical constructivist approach (see p. 306). Do the studies done in each case actually follow the approach to constructivism intended? Are the studies done from the same constructivist approach? Why the difference in constructivist approaches by the same author? (JE, 2/15)
How well does the model of Confrey (1990) fit Simon's claim that constructivism "provides a useful framework for thinking about mathematics learning in classrooms...[but] it does not stipulate a particular model" (p.114)? (JE, 2/16)
How do we teach students to use the process of reflective abstraction (Simon, et al., 2004; Dubinksy, 1991) or the reflective process (Confrey, 1990)? (JE, 2/16)
Adding to Jonathon's question above, how do we guide future teachers this reflective process? How can we equip them to reflect on their own teaching and learning? Especially in light of the fact that they are primarily learning through the lecture method, how best can we introduce them to constructivist ideas without overwhelming them (and ourselves) with so many approaches? (CZ, 2/16)
I had some questions about theory building. They're not gelling well for me, but here they are. What is the desired level of specificity in a theory of teaching? This week we saw some theories that carry implications for the daily practice of teaching and these results do not seem content specific. I could think about HLT as I design and carry out instruction for a differential equations course or for a grade three classroom. Are there possible gains in looking at HLT within specific content domains? For example, Sfard and Linchevski (1994) applied their theory of reification to algebra. (RK)
-RK- I was thinking about similar questions of this nature. Some of the authors we have read did not do much specific work in mathematics education (such as Piaget), while others have worked extensively in it (such as von Glasersfeld). In broader terms of constructivism, I wonder whether it is used and applied in other disciplines. I can easily see the similarities between problem-based mathematics and inquiry-based science classrooms. Similarities are less clear between mathematics, and say, history or foreign languages. With the idea that constructivism is centered around the constructing of new knowledge (whether this is cognitive, ie radical constructivist, or social), the use in mathematics classrooms seems centered around problem solving mathematics. With this being the case, I feel like constructivist mathematics classrooms are built around problem solving, not direct instruction. In history or foreign language classrooms, it doesn't seem like students are "problem solving" as it were, as much as they are assimilating new knowlege/vocabulary, and using that knowledge to analyze situations historically or to communicate. Without knowing more about how constructivism is used in other disciplines all together, it seems as if if we consider constructivist approaches those that encourage students to construct their knowledge, then mathematics classrooms also contain students constructing their knowledge, albeit from a direct instruction approach. Perhaps we need to revisit in class some of the deeper implications of constructivism. (JE, 2/16)
Questions about level of specificity also come to my mind when I think about the amount of prescription in a theory. Clearly the readings do not support a theory of teaching that prescribes moment-to-moment actions by the teacher (teaching is to be problematized, and so the teacher must be left room to make some of her own decisions). But one also wouldn't want a theory of teaching that is too vague. Balancing this inherent tension seems an important part of theory-building and I wonder how this weeks theories look like through this lens. (RK)
Rob, Rob, Rob, Rob, Rob - have you heard nothing about Assessment for Learning in the past year! (friendly jab, friendly jab). In response to your "clearly argument" I would point you to Simon (1995), p. 138, where he says the following: "the teacher is continually engaged in adjusting the learning trajectory that he [she] has hypothesized to better reflect enhanced knowledge." If this does not entail "moment-to-moment" (see Classroom Assessment Minute by Minute, Day by Day, by Leahy, Lyon, Thompson, and Wiliam, 2005) actions, what does? (JE, 2/16)
JE - I was not clear, I see. By "prescribes moment-to-moment actions" I mean rigid descriptions of teacher actions that span the time from the moment the teacher walks in the door to the moment they walk out. "First, you should do this. Then this. Etc." As an example of what I do not mean, I would point to the Launch-Explore-Summarize sequence of Core-Plus. This is presription for a sequence of activities, but there is quite a bit of room for variability within each activity. The quote that you point to, in which "the teacher is continually engaged in adjusting the learning trajectory that he [she] has hypothesized to better reflect enhanced knowledge" (p. 138) says little about how the teacher is engaged in making adjustments; that is, how the teacher solves the problems that arise in the course of following the learning trajectory. This, I believe, is left largely on the level of the teacher, who is to be an active problem-solver in the classroom. Perhaps your Leahy, Lyon, Thompson, and William (2005) reference contains some suggestions that can guide this problem-solving activity, but alas I have not yet read it. (RK)
Can I ask, too, what kind of differences do we see between theories proposed by constructivists and those proposed by logical positivists? Sometimes this week I felt as though I was reading about "how students really learn" in a way that seemed almost to acknowledge the presence and influence of an external reality. How do constructivists balance their skepticism with the need to put forth "global" theories of teaching and learning? Would a constructivist also support "local" theory building (on the level of individual teachers) through the publication of qualitative studies such as narratives? (RK)
As I was reading these past three weeks about constructivism, I kept thinking back to how I learned as a student and I thought that there were some concepts that did not produce a "learning perturbation" for me such as definitions or postulates. Does this mean that these concepts are not worth teaching because it may be hard to "learn" (as I did not construct it)? Or am I thinking about this all wrong? (JH)
-JH- I think back to a situation in teaching about right triangles, and I "perturbed" the students by looking at a "right triangle" on a globe, eventually getting them to see that we can have a triangle with three 90-degree angles. I think a perturbation is a situation where a student's current thinking is jeopardized and they have to decide how to adapt that information. (JE, 2/16)
This might be related to Janice's question directly above, but What exactly does it mean to be an active learner? (See Simon, 1995, p. 118). (NF)
-NF- See Phillips (1995), p. 9, right column for a football analogy of what active learning means (I swear, I didn't know of this article when we had this discussion a few weeks ago!) (JE, 2/16) -- In general, sports analogies don't resonate with me. Maybe that's why I glossed over it the first time... NF
Simon (1995) (among others) claims that conceptual development can be theoretically explained by Piaget’s notion of reflective abstraction. In reflecting on my own teaching experiences, I have difficulties in finding experiential evidence of conceptual knowledge development occurring. Any ideas on how to determine whether “the lesson fostered the development of conceptual understanding? Simon (1995) suggests that when students overcome obstacles that this provides evidence of conceptual growth (p. 139). What else? (NF)
-NF- I think to my own learning experiences in that the "challenge to body and mind" (Simon, 1995, p. 139) spurred me to grow mathematically. In fact, most growth came about this way. In other words, without cognitive conflict (or perturbations) growth was much more difficult to come by, even in classrooms dominated by direct instruction. To me, this comes back to the adaptation (assimilation or accommodation) of Piaget. (JE, 2/16)
Might a constructivist say that all humans engage in von Glasersfeld’s first and second principles of constructivism but not all necessarily believe in this perspective? (Inspired by Steffe & D’Ambrosio, 1995, p. 146). In a related vein, must both of von Glasersfeld’s principles be satisfied in order for one to instruct from a constructivist orientation? (Principles: “knowledge is not passively received but is actively built up by the cognizing subject”; “the function of cognition is adapted and serves in the organization of the experiential world rather than in the discovery of ontological reality” (von Glasersfeld, 1989, p. 162).) (NF)
-NF- If all humans engage in von Glasersfeld's two principles, then how do we distinguish between direct instruction and the theoretical implications of constructivism to teaching (Confrey, 1990)? (JE, 2/16)
Please help me remember if we have classified Vygotsky as a constructivist or not---Phillips classified him as such (Phillips, 1995, p. 7 first column) (JH).
-JH- I think of Vygotsky as a social constructivist. (JE, 2/16)
Confrey makes the statement (pg 111) : "To a constructivist, knowledge without belief is contradictory. Thus I wish to assert that personal autonomy is the backbone of the process of construction." Could someone help me understand how the second statement follows from the first. I do not see the relationship between them. (JDS)
-JDS- Perhaps a concrete example of knowledge without belief would be beneficial. Think of an elementary student who sees two equal-sized glasses of water, and is asked if they have the same amount of liquid (which they will respond to with a yes). When the water in one glass is transferred to a glass of a different size (in front of the student), and then asked the same question, they will respond that one glass has more water in it. This is a case where they will have knowledge that conflicts with belief. Once this conflict is discovered by the teacher (call it a misconception if you will), the teacher can then work with the student toward a better understanding. Hope this helps. (JE, 2/16)
Confrey also has a list of characterisations of strong constructions but states that the list is not exhaustive. Do we think it is possible to create an exhaustive list? In other words can we come up with a rubric for our students constructions? (I realise under the assumptions of constructivism one can never assess the constructs of another directly, but just as a thought exercise, would it be possible to make a rubric that would characterise a perfect construction?) (JDS)
-JDS- In order to construct the rubric, we would first have to agree on what the exemplary (perfect) construction would look like. I think this would be a challenge. (JE, 2/16)